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ANRV400-FL42-17 ARI 13 November 2009 14:22

Turbulent Plumes in NatureAndrew W. WoodsBP Institute, University of Cambridge, Cambridge CB3 OEZ, United Kingdom;email: [email protected]

Annu. Rev. Fluid Mech. 2010. 42:391–412

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev-fluid-121108-145430

Copyright c© 2010 by Annual Reviews.All rights reserved

0066-4189/10/0115-0391$20.00

Key Words

volcanic eruption, geostrophy, hydrothermal plumes, lake bubble plumes,sedimentation

AbstractThis review describes a range of natural processes leading to the formationof turbulent buoyant plumes, largely relating to volcanic processes, in whichthere are localized, intense releases of energy. Phenomena include volcaniceruption columns, bubble plumes in lakes, hydrothermal plumes, and plumesbeneath the ice in polar oceans. We assess how the dynamics is affected byheat transfer, particle fallout and recycling, and Earth’s rotation, as well asexplore some of the mixing of the ambient fluid produced by plumes in aconfined geometry.

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1. INTRODUCTION

In this review, we describe the dynamics of a number of geological and environmental processesthat lead to the formation of turbulent buoyant plumes, many of which involve multiphase flows.Turbulent buoyant plumes arise in flows with Reynolds numbers in excess of order 104 and canlead to substantial mixing between the plume and environmental fluid, as well as the injection anddispersal of material into the ambient fluid. One objective of models for such processes is to provideconstraints on the underlying geological or environmental phenomena; this leads to some interest-ing inverse problems in which one can use field observations of plume height and size or dispersalpatterns of solid ejecta to infer the strength of volcanic eruptions, hydrothermal plumes, or otherphenomena (Huppert 2002, Turner 1979). There is also interest concerning the ongoing mixingof the environment caused by such plumes when the ambient fluid is confined (Linden 2002).

The original analysis of turbulent buoyant convection from a maintained localized sourcereported by Morton et al. (1956) was a milestone in turbulence modeling. They recognized that,by averaging over a timescale long compared with the eddy turnover time and using horizontallyaveraged mass, Q, momentum, M, and buoyancy, B fluxes, a quasi-steady description of the plumedynamics could be developed in which

Q = 2π

∫rρvdr, M = 2π

∫rρv2dr, B = 2π

∫rg

(ρ − ρe )ρo

vdr, (1)

where v(r, z) and ρ(r, z) are the plume velocity and density, respectively; ρe (z) is the ambientdensity; and ρo is a reference density. In the limit that the changes in density are small comparedwith the background density, conservation of mass can be approximated by conservation of volume,and the density changes are only important in determining the buoyancy force. The plume motioncan then be described in terms of the specific volume, momentum, and buoyancy fluxes (q, m, andf ) defined as

q = b2u, m = b2u2, f = g ′b2u, (2)

in terms of the effective mean radius b, upward speed u, and buoyancy g′ of the plume. In suchhigh–Reynolds number turbulent plumes, the volume flux increases with height as a result of theentrainment and mixing of the ambient fluid by the plume. The detailed process of entrainmentinvolves the engulfment of finite parcels of the ambient fluid that are then incorporated intothe plume by mixing, as eddies overturn within the plume. However, on timescales that are longcompared with the eddy turnover time, the entrainment process can be modeled in terms of a meanhorizontal inflow velocity of the surrounding fluid that is proportional to the vertical speed in theplume, u. Conservation of mass, momentum, and bouyancy then leads to the model equations forthe plume

dqd z

= 2εm1/2, mdmd z

= g ′q 2,d fd z

= −N2q , (3)

as originally presented by Morton et al. (1956), where the ambient stratification is given byN2 = − g

ρe

dρed z . These equations have a similarity solution for the motion of a plume in a uni-

form environment (N = 0) in which the buoyancy flux remains constant and equal to the sourceflux fo and

q =(

6ε

5

) (9ε

10

)1/3

f 1/3o z5/3, m =

(9ε fo

10

)2/3

z4/3. (4)

The entrainment coefficient ε = 0.1 ± 0.01 has been determined by comparison with experiment(see List 1979, Morton et al. 1956). In a uniformly stratified environment (N = const.), buoyancy

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Maximum heightof rise of plume

Laterally spreadingintrusion

Ascendingturbulentplume withentrainment ofambient fluid

Figure 1Schematic of a turbulent buoyant plume.

flux in the plume decreases with height, which reduces the rate of increase of momentum flux withheight. Because the plume continues to entrain fluid, the mass flux increases, while the velocitydecreases. Eventually, the buoyancy flux falls to zero at the neutral buoyancy height, and the plumethen overshoots to the top height at which the speed falls to zero (Figure 1).

The fluid then slumps back and spreads out to form a lateral intrusion around the neutral height.The height of rise of the plume in a stratified environment can be scaled with the buoyancy fluxand the stratification of the ambient fluid, as measured by the Brunt-Vaisalla frequency N, andusing the entrainment coefficient, integration of the equations then gives the approximate scalinglaw (Turner 1979)

HT = 2.7(π fo )1/4 N−3/4, (5)

whereas the neutral height of the plume has a height given by the approximate scaling law (Turner1979)

HN = 2.1(π fo )1/4 N−3/4. (6)

It also follows from dimensional analysis that the time of rise of fluid through the plume scalesas 1/N (Woods & Caulfield 1992), and plumes can be regarded as quasi-steady if they persist fortimes greater than this, which has typical values of order 100 s in the atmosphere and 10,000 s(a few hours) in the ocean. These classical results have been used to provide estimates of themagnitude of the buoyancy source of a number of natural plume-forming phenomena, essentiallyby measuring the plume height HT , and inferring from Equation 5 that

fo = (2.7)−4 H 4T N 3/π. (7)

If there is some uncertainty in the measurement of HT , then this is amplified in the estimate ofthe buoyancy flux fo.

2. NONLINEAR BUOYANCY VARIATIONS WITH MIXING

An important challenge in the application of plume models to real phenomena lies in the assump-tion of conservation of buoyancy. Conservation of buoyancy is in fact a derived relation, associatedwith the conservation of enthalpy, salt, particle load, or other physical quantity, in the limit thatthe density difference (and hence buoyancy) is inversely proportional to the dilution. As the plumebecomes progressively more dilute, this approximation typically becomes quite accurate, and theplume may be described in terms of a conserved effective buoyancy flux. This effective buoyancyflux can be related to the initial flux of a conserved quantity, such as heat or particle load, and is

www.annualreviews.org • Turbulent Plumes in Nature 393

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00.20.40.60.81.01.21.41.61.8

Log

(hei

gh

t, k

m)

Log (volume eruption rate, m3 s–1) 0 1 2 3 4 5 6

Figure 2The height of rise of some historic volcanic eruptions, compared to the scaling law (Equation 6) based onestimates of the buoyancy flux associated with the heat flux of the eruption. Data taken from Sparks et al.1997.

found by evaluating the buoyancy flux of an equivalent diluted flow for which the buoyancy thenvaries inversely with the dilution (see Sparks et al. 1997, Turner & Campbell 1987). For example,the height of rise of a volcanic plume can be modeled approximately by an equivalent plume ofair, 1◦C hotter than the ambient, with the same heat flux QH as the erupted material, and this hasbuoyancy flux fo = gQH/(ρCp)aTe, where g is the acceleration of gravity; ρa and Cpa are the densityand specific heat of the air, respectively; and Te is the ambient temperature at the source. Using thisrelation, Equation 5, and independent estimates of eruption heat flux derived from field observa-tions of the volume eruption rate of solids and thermal energy per unit volume, the modeled heightof rise compares well with the observed height of rise of some historic eruption plumes (Figure 2).

One example of considerable interest in nature concerns hot particle-laden plumes in air. Ifthe particle mass fraction in the plume is (1 − n), the density of the hot air in the plume is ρg,and the particle density is ρ s, then the density of the air-particle flow relative to the ambient air,of density ρa, is given by

�ρ

ρa= ρg

ρa

[1 − ρa

ρgn − ρa

ρs(1 − n)

ρgρs

(1 − n) + n

]. (8)

If the system is isothermal, so ρg = ρa, it follows that the density is greater than the air. However,if the particles are initially hot, and are sufficiently small that they thermally equilibrate with theair, then, in the limit ρs > (ρg , ρa ), the mixture may be less dense than the air if

nρa > ρg . (9)

As the gas mass fraction n increases by mixing more cold ambient air with the particles, thetemperature and hence air density in the mixture ρg decrease according to the conservation ofheat with the particles. If initially the particles are sufficiently hot, then with small n the mixture isdense relative to the air, but if the flow becomes progressively more diluted with external air, themixture may become less dense than the ambient air. Such a reversal of the buoyancy can occurin volcanic eruption columns, as we describe below.

3. MODEL OF AN ATMOSPHERIC VOLCANIC PLUME

During explosive volcanic eruptions, dense mixtures of ash and gas issue from the volcanic ventat a speed of 100–150 m s−1, constrained by the choked flow speed of the erupting mixtureissuing from the vent. This mixture then ascends into the atmosphere, where it decompressesto atmospheric pressure and ascends upward, initially driven by its momentum, entraining andmixing with ambient air as it ascends. This entrained air lowers the density of the mixture, and

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Vel

oci

ty o

f jet

(m s

–1)

Log (mass eruption rate, kg s–1)

300

250

200

150

100

50

06 7 8 9 10

a3%

5%

b

Figure 3(a) Photograph of the 15-km-high Mt. Redoubt eruption column, formed during the April 5, 1990, eruption.(b) Calculation of the critical eruption velocity required so that a buoyant plume can develop by entrainingand heating air. The two curves correspond to different magma water content. Data taken from Woods 1988.

if sufficient air is entrained (Equation 9), the mixture becomes less dense than the environment.At this point, the jet transforms into a buoyant plume and then continues to rise through theatmosphere, entraining more air and becoming progressively less dense until it reaches a minimumbuoyancy. Subsequently the density increases back toward that of the stably stratified ambient, andthe plume reaches a maximum height at which it intrudes laterally into the atmosphere (Sparkset al. 1997). Figure 3a shows a photograph of the Mt. Redoubt eruption cloud from the April 5,1990, eruption, which ascended 15 km into the atmosphere.

The rate of entrainment in the lower dense jet region is critical for determining whether theflow becomes buoyant before or after the initial momentum is dissipated. In the latter case adense fountain collapses, leading to dense ash flows that spread from the volcanic vent. Becausethe density of the particle-gas mixture varies nonmonotonically relative to the ambient density, indeveloping a model of the motion of such plumes, it is necessary to include the density explicitly inthe model and hence work in terms of the mass, momentum, particle, and enthalpy fluxes, leadingto (see Woods 1988)

d Qd z

= 2ερe ub,d Md z

= g(ρe − ρ)b2,d Q(1 − n)

d z= 0,

dc p (T − To )Qd z

= 2ερe ubc p (Te − To ),(10)

where n is the gas mass fraction; T(z), Te(z), and To denote the mixture temperature, the ambienttemperature, and a reference temperature, respectively; and ρe denotes the ambient density.Equation 10 is combined with the equation for the density of the particle-air mixture (Equation 8)and a model of the atmospheric density profile to account for changes in pressure with height(Woods 1988). We note that the far-right relation in Equation 10 is a simplified enthalpy equationwritten in terms of the conservation of internal energy (Woods 1988). The entrainment coefficientε varies depending on whether the flow has positive or negative buoyancy (Kaminski et al. 2005,Suzuki et al. 2005, Turner 1986), with a pure jet having a value ∼0.06 ± 0.01 and a buoyant plumehaving a value ∼0.1 ± 0.01. Suzuki et al. (2005) suggest from full numerical simulations that avalue of approximately 0.07 may be appropriate near the conditions for collapse. With large densitycontrasts and high source pressure, the near-source entrainment also depends on the density andpressure ratio, and this is less well understood (Ricou & Spalding 1961, Thring & Newby 1953).


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Although somewhat simplified, a number of the basic principles concerning the flow evolutioncan be inferred by using a constant value for ε. One prediction is that there is a critical eruptionspeed of the mixture, as a function of the eruption mass flux, for which the mixture becomesbuoyant through entrainment and heating of air (Figure 3b): Smaller fluxes lead to column collapseand dense ash flow formation. Woods & Caulfield (1992) and more recently Kaminski et al.(2005) modeled this balance between buoyancy generation by entrainment and the dissipationof momentum through an analog series of experiments using mixtures of methanol and ethyleneglycol (MEG) with water; the density of a MEG-water mixture varies nonmonotonically relativeto water, with MEG-rich mixtures being less dense than water, but becoming more dense thanwater as the water content increases. These experiments demonstrated that one can describe thetransition from plume to pyroclastic flow through a version of the above plume model adaptedfor the analog experimental system.

In the homogeneous model above, a critical assumption is the thermal and dynamic equilibra-tion between the particles and the air, so that the bulk temperature of the particle-air mixture isgiven by the conservation of heat, and the slip velocity between the particles and air is assumed tobe small compared to the bulk rise speed. These assumptions require that the solid particles aresufficiently small. With a turbulent buoyant plume, the timescale of ascent in the plume througha stratified environment, ∼1/N, is of order 100 s in the atmosphere. The time for thermal equi-libration is of the order of τ τ ∼ d 2/κ , where d is the particle size, and κ is the thermal diffusivity.Therefore, for thermal equilibration, the particle size should be smaller than approximately 1 mm.The time for dynamic equilibration of small particles of radius d scales as τ d ∼ d 2/ν, where v is thekinematic viscosity of the air. This is small compared to the rise time if the particles are smallerthan about 1 mm. With larger particles, there may be some particle fallout and inefficient heattransfer in the ascending flow. This delays the development of buoyancy in the dense jet regionaround the vent, produces two-phase flows, and increases the critical eruption speed for whichmaterial erupting with a given flux can become buoyant and form a plume (Woods & Bursik 1991).

4. THE NEUTRAL CLOUD

4.1. Inertia: Buoyancy Dispersal

In the event that a turbulent buoyant plume does become established in an eruption of dominantlyfine ash, a laterally spreading intrusion develops above the plume at the neutral buoyancy height.In the initial spreading cloud, there is a balance between the buoyancy force and the fluid inertia,but the motion may eventually become dominated by wind forcing (Sparks et al. 1997) or, inlong-lived plumes, by the Coriolis force associated with Earth’s rotation (Bush & Woods 1999,Helfrich & Battisti 1995). In a stratified environment, the advancing current can also generateinternal waves in the ambient fluid (Flynn & Sutherland 2004), and near the intrusion, the ambientfluid is displaced to accommodate the continuing injection of plume fluid. A range of scaling lawshas been proposed for the spreading rate of an intrusion with a continuous source in a stratifiedambient (Chen 1980, Didden & Maxworthy 1982, Lemkert & Imberger 1993), and the results ofexperiments (e.g., Kotsovinos 2000) are not fully conclusive. Models of the spreading of a finiterelease of fluid in a stratified ambient suggest that the nose, of depth h, spreads radially with speed

u = �Nh, (11)

where � is a constant of order unity (Amen & Maxworthy 1980, Flynn & Sutherland 2004,Maxworthy et al. 2002). If we adopt this law, and assume that the flow is quasi-steady, so that atthe nose of the intrusion, the flux equals that supplied at the center of the intrusion, 2πρrhu = Q,

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then we obtain a power-law scaling for the radius (Chen 1980, Lemkert & Imberger 1993)

r =(�QN

8πρ

)1/3

t2/3. (12)

This scaling law was successfully compared with observations of the umbrella cloud during thefirst 5 h of the eruption of Mt. Pinatubo on June 15, 1991 (figure 8 in Holasek et al. 1996), duringwhich time the cloud radius increased to 400–500 km. Below we discuss how, on longer timescales,as the flow evolves, the effects of Earth’s rotation or of ambient winds may become dominant.

4.2. Particle Fallout

As the particle-gas mixture spreads laterally from the volcano according to Equation 12, theparticles begin to settle, leading to a deposit of ash around the volcano. We denote the volumefraction of particles in the spreading neutral cloud as ψ , and we equate the mass flux of fluidsupplied by the plume at the neutral height, Q(H ), with the outward flow in the neutral cloudQ(H ) = 2πρruh, where the radius of the spreading cloud is r, the outward speed is u, the depth ofthe flow is h, and ρe is the ambient density at the height of the intrusion H. Typically, in volcanicplumes, the neutral cloud is highly turbulent as it spreads outward, so it behaves as a well-mixedcurrent, with particles settling from the lower surface at a rate proportional to the particle load andthe fall speed (Hazen 1904). Although there is typically a range of particle sizes, for the presentreview it suffices to consider particles with the mean settling speed vs, noting that the model maybe readily extended to include effects of different particle sizes (see Sparks et al. 1997). The particlevolume fraction in the neutral cloud thus decreases with radius as

∂

∂r[ruhψ] = −rvs ψ, (13)

and the mass of particles in the neutral cloud decays with radius as

ψ = ψo exp[−πρvs (r2 − r2

o )Q(H )

], (14)

where ro is the outer radius of the plume at the neutral height. By combining the vertical fall speedof the particles with the radial inflow speed associated with the entrainment of air, we find thatsmall ash particles, which settle according to Stokes law, follow a trajectory (r, z) given by

drdt

= −εbur

,d zdt

= −vs . (15)

These relations suggest that the maximum radius in the neutral cloud at which particles settlingfrom the neutral cloud can become re-entrained into the plume is

r2m = Q(H ) − Q(0)

πρevs. (16)

If particles fallout at a radius r such that r < rm, particles may be re-entrained into the plume atheight z given implicitly by the relation

r2 = b2 + Q(H ) − Q(z)πρevs

. (17)

In contrast, particles that sediment from a point rs > rm fall out to the ground at radius rg givenby

r2g = r2

s − Q(H )πρevs

, (18)


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and the flux that sediments at radius rg per unit area is given from Equation 14 by

S(rg ) ∼ Q(H )2πrg

∂ψ

∂r

∣∣∣∣rs

. (19)

This law for the sediment pattern has been successfully compared with field observations froma number of axisymmetric volcanic eruption deposits, notably Fogo A in the Azores (Sparkset al. 1997), and also with laboratory experiments using particle-laden plumes of fresh waterrising through a tank filled with aqueous saline solution (Sparks et al. 1991). The model predictsdifferential settling of coarser material near the source and finer material further from the source; italso predicts that if eruption intensity increases (decreases), deposits have an increasing (decreasing)mean particle size at a given location. Further from the source, the transport processes becomedominated by zonal winds and the associated dispersive mixing (Sparks et al. 1997).

4.3. Collapse Through Particle Re-Entrainment

One effect of interest for the plume dynamics concerns the recycling of particles into the plume,which increases the plume’s density and, as a result, may lead to its collapse. Veitch & Woods(2000) developed experiments to examine the re-entrainment from the spreading neutral cloud.They injected fresh particle-laden water into the base of a tank of saline solution, with the particleload being sufficiently small that the fresh water led to the formation of a buoyant plume. Asparticles began to settle through the ambient fluid, some were gradually re-entrained into theplume, decreasing the buoyancy until, eventually, the mixture ceased to be buoyant and the plumecollapsed. Following the collapse, and the associated fallout of the particles from the environment,the plume was re-established as a transient flow until the re-entrainment of particles resumed,leading to a further collapse event.

By equating the total flux of solid entrained between the source and height z in the plume withthe flux of particles that fall out of the neutral cloud between radius r(z) and r(0) (Equation 17),we find that a steady state can become established with the volume flux of particles in the plumeat height z, Fp(z), and at the source, Fp(0), related by

Fp (z) = Fp (0) exp[

Q(z) − Q(0)Q(H )

], (20)

where Fp(0) is the source particle flux, and we have assumed that the plume radius b(H ) � rm, theradius on the neutral cloud at which the particles are just re-entrained into the plume (Equation 16;Veitch & Woods 2002). Typically the volume flux in a plume increases with height by several ordersof magnitude through entrainment prior to reaching the neutral point Q(0) � Q(H ), so themaximum additional particle load in the plume, owing to re-entrainment, is given approximatelyby the relation

Fp (H ) ∼ e Fp (0). (21)

This additional particle load is entrained over the vertical extent of the plume and thereforeincreases with height in the plume. To determine whether plume collapse can be induced throughre-entrainment, one can find the buoyancy of the plume at each height above the source, comparingthe predictions of the plume model prior to the establishment of the recycling and the predictionsof the model including the recycling process. We note that a quasi-steady plume, with no significantrecycling, becomes established in a time of order 1/N ∼ 100 s, while the particle recycling, andassociated formation of a steady plume that includes the dynamic impact of the recycling, developsover a time of order H/vs ∼ 104–105 s for particles with fall speed of 0.1–1 m s−1 in a 10–20-km-highplume.

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a b c d e f

Figure 4Series of photographs from Veitch & Woods’s (2000) particle-recycling experiments, illustrating the periodic collapse of the plumethrough particle re-entrainment.

The bulk density of the experimental water-particle plumes, relative to the saline water, �ρ, isgiven by

�ρ = − (ρp − ρw

) Fp (z)Q(z)

+ (ρs − ρw)Q(0) − Fp (0)

Q(z), (22)

where the first term denotes the negative buoyancy of the particles, and the second term denotes thepositive buoyancy of the fresh water. Once the plume is established (but before the re-entrainmentprocess has developed), Fp(z) is independent of z, and Equation 22 gives a condition that the flowis buoyant, shown by the first inequality in Equation 23; however, once the recycling process isestablished, so that Fp(z) increases with height according to Equation 21, then the plume maycease to be buoyant before reaching height H, if the second inequality in Equation 23 applies:

Fp (0)(ρp − ρw

)<

[Q(0) − Fp (0)

](ρs − ρw) < e Fp (0)

(ρp − ρw

). (23)

As well as testing this simplified picture of the re-entrainment with a series of laboratory exper-iments (Figure 4), Veitch & Woods (2000, 2002) also applied the model to volcanic eruptionplumes, accounting for the re-entrainment in addition to the other effects discussed above. Theyillustrated that the process of particle recycling can indeed lead to the collapse of an eruptioncolumn and hence cycling between flow and fall behavior in the deposits.

4.4. The Neutral Cloud: Geostrophic Adjustment

As mentioned above, in long-lived eruptions, Earth’s rotation can impact the neutral cloud dy-namics. These effects were first studied in the context of hydrothermal plumes, which are plumesof hot water and minerals that issue from the seabed, and we consider that application below.First, we describe the dynamics of rotating neutral clouds, building from the experimental andtheoretical work of Helfrich & Battisti (1995) and Woods & Bush (1999).

As a neutral cloud spreads through a stratified ambient, the initial dynamics is controlled bythe buoyancy-inertia balance described above (Section 4.1). However, as the fluid spreads radially,it develops an anticyclonic circulation with azimuthal speed (Gill 1981)

vs ∼ ωr (24)

at radius r from the plume top, essentially to conserve angular momentum, where 2ω representsthe vertical component of Earth’s rotation at the latitude of the plume. (In texts on rotating flows,this is normally denoted by f, but here we use ω to avoid any confusion with the buoyancy flux.)The Coriolis force associated with this swirling flow gradually begins to balance the outward radialbuoyancy force associated with the background stratification as expressed by the constant value ofthe Prandtl ratio,

P = NH/ωR ≈ 0.5 ± 0.2, (25)

where the half-height of the neutral cloud is H, and the radius is R, whereas the constant 0.5 ±0.2 has been determined from laboratory experiments (Bush & Woods 1999, Helfrich & Battisti


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1995). The volume of the spreading geostrophic neutral cloud has been measured to follow the lawV = 2πβHR2, where β ∼ 2.7 is an empirical shape factor. Combining this with the geostrophicconstraint (Equation 25), we find with a constant mass supply Q(H ) from a plume, the position ofthe outer edge of the neutral cloud, R(t), increases with time according to the relation

R(t) ≈(

QN2πρωβBg

)1/3

t1/3. (26)

Comparing this with Equation 12 suggests a transition in spreading regime from the inertialbalance to the geostrophic balance after a time of order

T ∼ 1/ω. (27)

If an eruption does not persist this long, then the neutral cloud is emplaced according to theinertia-buoyancy balance, but in the absence of strong zonal winds, then over a time T, the cloudadjusts to geostrophic balance with a radius given in terms of the cloud volume V as

R(t) ≈(

V N2πωβBg

)1/3

. (28)

For example, in an eruption that rises to a height of approximately 20 km, the plume suppliesa volume flux of approximately 1010 m3 s−1 to the neutral cloud; if the eruption persists for aperiod of 103–104 s, then the geostrophically adjusted neutral cloud has a radius of 50–100 km,comparable with the initial radius of the cloud predicted by the simple inertia-buoyancy balance(see Baines & Sparks 2005).

Satellite images of the top surface of the neutral cloud produced during the June 15, 1991,eruption of Mt. Pinatubo suggest that, although the initial spreading was controlled by an inertia-buoyancy balance (figure 8 in Holasek et al. 1996), after a time of 8–10 h, a series of eddy-typefeatures developed on the neutral cloud, implying the development of an anticyclonic swirlingflow.

However, given that the zonal winds in the atmosphere have speeds up to 10 m s−1, we expectthat in smaller eruptions, once the neutral clouds reach scales of order 100 km, their spreadingrates (Equation 12 and 26) will fall below that of the zonal winds, and the cloud will be carrieddownwind (Sparks et al. 1997).

5. SUBMARINE PUMICE PLUMES

When a mixture of hot pumice and ash erupts through a submarine vent, there is an initial highlyexplosive interaction in which the sea water quenches the outer surface of the erupting material,producing a hot jet of pumice, ash, and water, with the vapor phase that erupts from the volcanocondensing as it mixes with the sea water (Head & Wilson 2003, Kano et al. 1996, Woods 2009).The density of this mixture is then affected by the buoyancy of the hot water and the buoyancy ofthe pumice clasts. The pumice clasts, which have typical sizes in the range 0.01–1 cm, are filledwith vapor bubbles that form as the magma decompresses during ascent through the crust to thesurface. If the bubble volume fraction is sufficient, the pumice clasts may be less dense than thesurrounding water; the solid material in the pumice typically has a density of order 2500 kg m−3

compared with water (density of order 1000 kg m−3), whereas the water-vapor density dependson the depth of water and temperature, but is of order 1–20 kg m−3 in water of depth <1000 mand with temperatures up to approximately 1000◦C (Allen et al. 2008). Depending on the depth,if the bubble volume fraction of the pumice is approximately 0.6 or greater, then the pumice isbuoyant (Figure 5). This, together with the heating of the water (which cools the pumice and

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Collapsing flow aspumice cool and become water loggedand dense

Near-surface current with some large buoyant pumice, whicheventually cool, become dense, and sink back to the seabed

Dense pumice sediments to

form flow deposit

Submarine eruption withinitially buoyant pumice

forming buoyant flow

Figure 5Schematic of a submarine pumice eruption, showing the transition to a collapsing fountain as the smallerpumice collapse and become water logged and dense.

fine ash), may initially promote the formation of a buoyant plume. If the density contrast betweenthe pumice and the water is 0.1 times the density of the water, then the rise speed of individualparticles is approximately 0.01–0.1 m s−1, whereas a group of such particles in suspension with avolume concentration of order 0.1, and lateral scale of 10–50 m, would lead to a bulk upflow speedof order 1.0 m s−1, suggesting that the pumice rise convectively with the fluid and could ascend100 m in times of order 100 s.

Submarine explosive eruptions can therefore produce pumice plumes driven by the buoyancy ofthe pumice clasts, as well as the thermal expansion of the water. However, many field observationsof deposits suggest that these plumes collapse back to the seafloor to form large density currents.This is a result of the cooling of the pumice clasts as they rise through the relatively cold watercolumn. On cooling, the vapor in the bubbles is replaced by water, and the pumice clasts thereforebecome relatively dense. The cooling occurs over a conductive time that scales with the particlesize and the thermal conductivity. For pumice particles of size 0.1–1.0 cm, this cooling time is oforder 10–1000 s, comparable to the ascent time to the surface. Depending on the size distributionof the pumice clasts, the bulk density of the flow may then increase above that of the water,producing a negatively buoyant jet, and the flow is then arrested and collapses back down to theseafloor (Figure 5). At the top of the fountain, the larger pumice particles, which may be buoyant,may separate from the bulk mixture and, together with some of the warm water, form a smallerascending plume, whereas the main flow collapses back to the seafloor to form a dense sedimentingflow (Allen et al. 2008, Woods 2009).

6. HYDROTHERMAL PLUMES

6.1. Heat Flux Estimate

In the deep ocean, hydrothermal plumes develop as hot mineral-rich water, associated with thelarge convective cooling of the new ocean crust at mid-ocean ridges, issues from the seafloor.The plumes ascend through the weakly stratified deep ocean and reach heights up to 200–300 mabove the sea floor (Speer & Rona 1989). The heat content of hydrothermal plumes is relevantfor constraining models of the heat budget of Earth. However, to estimate the heat flux from the


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height of rise of the plume, using plume theory, one must consider both the solutal and thermalstratification of the ocean. The salt and heat fluxes are given in terms of the specific volume flux (q);the salt concentration (mass/volume) and temperature in the plume [S(z), T(z)] and the ambient[Se(z), Te(z)],

Qs (z) = π [S(z) − Se (0)] q (z), QT(z) = πρCp [T(z) − Te (0)] q (z); (29)

together with an expression for the buoyancy of the plume, which for small differences in tem-perature and salinity can be linearized to the form

g ′ = [ρ(z) − ρe (z)]ρe (0)

= g {β [S(z) − Se (z)] − α [T(z) − Te (z)]} , (30)

where α and β are the thermal and solutal expansion coefficients, respectively. In general, withhigh initial temperatures, the thermal expansion coefficient is nonlinear, so although the heat fluxis conserved, the buoyancy flux varies with mixing even in a uniform environment (see Turner &Campbell 1987). To draw on the classical plume results (which apply when the density differenceis linearly proportional to the temperature difference), we therefore work with an analog diluteplume comprising plume fluid mixed with sea-bottom water so as to preserve the total heat and saltfluxes, but for which, on mixing, the buoyancy varies linearly with the temperature and salinity.The governing equations are then identical to Equation 3, but the buoyancy now comprisescomponents from both heat and salt, and the Brunt-Vaisalla frequency is given by the sum of thetemperature and salinity gradient:

N2 = N2T + N2

S = αd Te

d z− β

d Se

d z. (31)

The height of rise of the plume is then given by the classical plume model, H = 2.7 [π f (0)]1/4N−3/4

(Equation 5), where f(0) denotes the bulk buoyancy flux accounting for both the heat and salinityfluxes. To determine the heat flux associated with the plume, we separate the thermal and solutalparts of the buoyancy flux. If the salinity and potential temperature gradients are linear, then thesalt flux Qs(z) and the heat flux QT (z) may be expressed in terms of the specific volume flux q(z) as

Qs (z) = Qs (0) + πd Se

d z

z∫0

dqd z

zd z, QT(z) = QT(0) + πρCpd Te

d z

z∫0

dqd z

zd z, (32)

whereas the variation of the buoyancy flux of the plume with height leads to the relationz∫

0

dqd z

zd z = q (z)z + f (z) − f (0)N2

. (33)

Combining these relations with the definition of the heat flux, and noting that at the neutral heightf (H ) = 0, we can estimate the total heat flux associated with a hydrothermal plume by

QT(0) = πρCp

[d Te

d zf (0)N2

+ q (z) [T(H ) − Te (H )]]

, (34)

where the total buoyancy flux at the origin f(0) = H4N3/π (2.7)4, and we measure the temperatureanomaly of the neutral cloud relative to the ambient at that height.

6.2. Instability of Hydrothermal Plumes

Hydrothermal plumes are typically long-lived, so the dynamics of the neutral cloud tends togrow in geostrophic equilibrium. However, as the neutral cloud continues to grow and deepen,

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it eventually becomes unstable as the inward velocity field associated with the entrainment fielddevelops a cyclonic circulation, and the neutral cloud’s thickness becomes comparable to its height(Helfrich & Battisti 1995). The instability occurs over a time of order 100 N/ω2, which in thedeep ocean corresponds to a time of order 10–100 days. The neutral cloud is then shed from thesource as an anticyclonic vortex with radius

R = 2.5 f (0)1/4 N1/4ω−1 (35)

in terms of the source buoyancy flux of the plume, f(0). This eddy, for which R ∼ 10H, may then becarried away from the source by any weak currents, and a new neutral cloud becomes establishedabove the plume.

6.3. Seafloor Lava Eruptions and Line Plumes

At mid-ocean ridges, there are also long fissures that open and erupt lava onto the seafloor, whichleads to very rapid heat transfer from solidifying basaltic lava into the overlying water column.With a long line source, a two-dimensional line plume forms. The hot plume fluid ascends andmixes until it reaches its neutral height, at which it then spreads laterally. However, owing to theCoriolis force, it then develops a circulation and becomes unstable, eventually breaking up intoa series of vortices, again with a Prandtl ratio P = Nh/ωR of approximately 0.5, where h is thehalf-depth and R is the radius of the geostrophic eddy (Figure 6; Bush & Woods 1999). Thevolume of one such eddy of radius R is then given by 2πβR2h, with β � 2.7, and using the Prandtlratio constraint, this can be related to the volume per unit length of the neutral cloud VL and theradius R of the eddy according to the relation

R =(

2NVL

βωP

)1/2

. (36)

Using this result (together with expressions for the volume flux supplied to the neutral cloud perunit length of the source, and for the height of rise of the line plume), Woods & Bush (1999)developed an analogous expression to Equation 35 to determine the heat flux contained in the so-called megaplumes that are thought to be produced from such events. In particular, they estimatedthat two large eddies that ascended approximately 700 m into the water column and extended 8–9 km in radius, found near the Juan de Fuca ridge (Baker et al. 1989), had an associated heat fluxof order 3.5–7 × 1016 J.

Figure 6Illustration of a neutral cloud above a line plume undergoing instability into a series of geostrophic eddies.Figure taken from Bush & Woods 1999.


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7. CO2 BUBBLE PLUMES AND LAKE ERUPTIONS

We now turn to another phenomenon occurring in nature, namely, lake eruptions of CO2, suchas those that occurred at Lake Nyos, Cameroon, in 1986 (Freeth 1987) and Lake Monoun,Cameroon, in 1984 (Sigurdsson et al. 1987). In these lakes, there is a small flux of magmaticCO2 supplied to the lake bottom. Because CO2 is soluble in water (and its solubility increases withpressure), the lake-bottom waters therefore gradually become charged with CO2. While the CO2

remains in solution, this process leads to a stable stratification of the lake, as the water densityincreases with dissolved CO2 content. However, eventually, if the lake-bottom waters becomesaturated in CO2, then any vertical mixing of the lake bottom water can lead to the exsolution ofbubbles. This may lead to convective overturn because the bulk density of the bubbly mixture mayfall beneath that of the surrounding water. Interest in such overturn events has been rekindled bythe ongoing risk assessment of possible CO2 sequestration projects.

A number of models have been proposed about the precise mechanism leading to an eruption,but the key process required is that the water near the lake bottom rises and decompresses,thereby releasing CO2 gas bubbles. Gas bubbles of order 0.01–0.1 mm in diameter have a risespeed through the water of order 0.01–1.0 cm s−1. If the convective rise speed of a group of bubblesand water exceeds this, then we expect a large-scale bulk convective flow rather than the ascent ofindividual bubbles. Woods & Phillips (1999) examined the possible ascent of a turbulent bubbleplume, supplied by the slow recharge flux at the base of the lake, assuming that the bubble-watermix rose as a homogeneous phase. Motivated by some field observations, they assumed that theplume developed when the lake-bottom water became saturated in CO2. Taking the gradient ofCO2 dissolved in the lake water adjacent to the lake bottom to have a constant fraction (<1) ofthe saturation gradient, they modeled the motion in terms of the conservation laws for mass flux(Q), momentum flux (M), and CO2 flux (C),

d Qd z

= 2ερ1/2e M1/2, M

d Md z

= g(ρe − ρ)

ρeQ2,

dCd z

= 2ερ1/2e M1/2ne , (37)

where ρe is the ambient density, and ρw is the density of the water in the plume, together with anequation for the density of the bubbles (ρb) and of the bubbly water mix (ρ):

ρb = P (z)RT

, ρ =[

nb

ρb+ 1 − nb

ρw

]−1

, (38)

with nb denoting the mass fraction of exsolved bubbles, given by nb = max(n − ns, 0), where ns isthe mass fraction of CO2 in the water, which is just saturated and varies with pressure. The totalmass fraction of CO2 in the plume is given by n = C/Q, and ne is the mass fraction of CO2 in theambient.

These equations predict that the bubble-plume mixture accelerates rapidly from the lake bot-tom, generating momentum. As it ascends, it mixes with water higher in the lake that is progres-sively more undersaturated in CO2, and eventually the small CO2 bubbles may be resorbed intothe water, reducing the buoyancy. The flow is then driven upward by its momentum through thestably stratified lake, and the plume decelerates (Figure 7). Calculations show that if the initialCO2 flux is sufficiently large, then the momentum generated near the lake bottom is able to carrythe plume up to the lake surface; at the low, near-surface pressures, the dissolved CO2 is thenreleased from the rising water and vents into the atmosphere. In contrast, with smaller initial CO2

flux (and hence plume momentum), the plume stops within the lower 50–100 m, where it spreadsout as an intrusion, with no release of CO2 to the atmosphere (Figure 8). The model predictsthat as the CO2 concentration gradient in the lake increases, the critical source flux for a surfaceeruption becomes smaller and eventually coincides with the estimates of the natural recharge flux

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CO2 concentration

Hei

gh

t ab

ove

lake

bed

Saturation value

Lake value

Upward motiongenerates gas

Mixing unsaturated water resorbs gas

Water just saturated at the lake bed

Gas jet issues from surface

Near-surface gas exsolves as pressure falls

Plume

For plume to reach surface need critical mass of CO2 in base of lake

Figure 7Schematic of the phenomena driving the formation of a CO2-bubble plume through a lake saturated in CO2.The schematic on the right illustrates the variation of the dissolved CO2 concentration of the plume ascompared to the CO2 concentration of the lake and the saturation CO2 concentration, as a function ofheight in the lake. In equilibrium, the bubbles are predicted to redissolve in the mid-depths of the lake, asmore undersaturated lake water is entrained.

on the lake bed. It is therefore possible that an overturning event can be triggered by a smalldisturbance in the source CO2 flux, as the lake CO2 concentration approaches the critical gradientfor eruption given the natural CO2 recharge flux. Also, the model predicts that the CO2 flux atthe lake surface is several orders of magnitude greater than that at the lake floor owing to theentrainment of the CO2-rich lake water.

However, numerous authors have proposed other trigger mechanisms, including the possibilityof a landslide or wind-driven seiche developing in the lake (Sigurdsson et al. 1987, Zhang & Kling2006). To evaluate these different trigger mechanisms, Mott & Woods (2009b) recently developed

Source CO2 flux (kg s–1)

Hei

gh

t of r

ise

(m)

0.0001 0.001 0.01 0.1 1 10

200

150

100

50

0

1.5

1.75 2.0 5.0

Lake surface

Figure 8Calculation of the height of rise of the bubble plume as a function of the lake-bottom CO2 flux for threedifferent values of the CO2 undersaturation of the lake, measured as the ratio of the actual CO2 gradient inthe lower part of the model lake to the saturation gradient of CO2, assuming the lake bottom is just saturated.


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a model to explore whether a mixing event that homogenizes the bottom few meters of the lakewater could lead to the ascent of a series of discrete CO2-laden thermals from the lake bottom. Ifthe CO2 concentration gradient is sufficiently close to the saturation gradient, then the mixing atthe lake bottom can lead to the release of bubbles, and this lower layer of bubble-laden buoyantfluid may become unstable and ascend as a series of discrete thermals.

Mott & Woods (2009b) have modeled the ascent of such discrete thermal clouds by writingconservation relations for the total mass, momentum, and CO2 content of a discrete thermal cloud(see Scorer 1957):

d QT

dt= εTuQ2/3

T ρ1/3e ,

d MT

dt= 2

3g

(ρe − ρ)ρe

QT,dCT

dt= εTuQ2/3

T ρ1/3e ne , (39)

where QT is the mass, MT is the momentum, and CT is the CO2 content. In the model, theinitial thermal consists of a well-mixed parcel of lake-bottom water, of radius h. The entrainmentparameter for the thermal, εT , has a value of approximately 0.33 (Scorer 1957). For reasonablevalues of the CO2 concentration gradient, a mixed layer of 3–4 m is sufficient for the thermalto reach the surface. These calculations suggest that each thermal may carry approximately 105–106 kg of CO2 to the lake surface after entraining the deep CO2-rich lake water during ascent.The model suggests that if a spreading particle-laden gravity current produced by a landslide, orbottom boundary mixing produced by a seiching motion, generates 100–1000 thermals, each ofsize 25–50 m3 (perhaps by mixing a region 3–4 m deep and of area 100 m × 100 m above the lakebed), then this process could also enable the release of a large volume of CO2 from the lake bed.

7.1. Lake Kivu

Lake Kivu is much larger than either Lake Monoun or Lake Nyos, extending over 400 m in depth,and it contains both CO2 and methane in solution in the lower 100–200 m of the lake. Althoughthe dissolved gas content is much smaller than the saturation value, there has been concern thatthe fissure that supplies magma to Mt. Nyirogongo could propagate below the lake, leading to alava eruption on the lake floor. This might then cause a hazardous release of CO2 from the lake.Woods & Baxter (2004) developed a simplified model of this process, accounting for the releaseof heat and CO2 as a lava flow erupted on the lake bed. They determined that there is a criticalheat flux from the lava flow that a line plume can ascend from the fissure up to the lake surface.

The model predicts that for a vigorous basaltic fissure eruption, in which there may be a fluxof 0.5–1.0 wt% CO2 erupted with the lava, the relatively small mass of dissolved CO2 in the laketypically only leads to an increase in the CO2 flux at the lake surface by a factor of two to three timesthe flux released from the lava flow, unlike the case of Lake Nyos. However, one risk associatedwith the potential eruption at Lake Kivu is that the magmatic CO2 could be transported to the lakesurface, where it would issue at a temperature close to that of the lake. Cold CO2 is dense comparedwith the air and would therefore spread out over the lake surface, without significant dilution. Thisrisk does not arise when CO2 issues from a subaerial fissure eruption because the CO2 is very hotand rises as a plume that is rapidly diluted, thereby substantially mitigating any health risk.

8. PLUMES IN CONFINED SPACES

8.1. Mixing Across Density Interfaces

Turbulent buoyant plumes can act as a tremendous stirring agent, entraining and transporting fluidthrough an ambient stratification. In a large open domain, the action of the plume is primarily

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to transport fluid through the stratification. However, in an enclosed region, the entrainmentcan lead to significant return flows in the ambient, which change the stratification. The classicproblem of such mixing, often called the filling box, was developed by Baines & Turner (1969)and concerned the mixing of a uniform enclosed fluid by a dense plume supplied at the top of thespace. As the plume descends, an upward return flow develops. At the base of the tank, the plumefluid spreads out and then ascends through the interior, behind a front between the original fluidand the new plume fluid. The continuing plume becomes progressively more dense at the base ofthe tank, as it passes through the first front and entrains from the deepening layer of plume fluid.The return flow and the speed of the first front are given in terms of the local volume flux of theplume πq, and the cross-sectional area of the reservoir, A, assuming that this is substantially largerthan the area of the plume. Assuming diffusion is small, then the density behind the front evolvesaccording to the relation (Baines & Turner 1969)

∂ρ

∂t+ πq

A∂ρ

∂z= 0. (40)

However, in an oceanic or lake setting, there is often a density stratification, so turbulent buoyantplumes are influenced by the stratification and may even be arrested, leading to the formationof intrusions within the interior of the ambient fluid. Kumagai (1984) studied the mixing acrossa two-layer interface, and Cardoso & Woods (1993) studied the mixing of a linearly stratifiedfluid by a plume. Recently, Mott & Woods (2009a) carried out some more detailed experimentsrelating to the mixing of a stratified fluid by a turbulent buoyant plume. They showed that whenan intrusion develops within the ambient fluid, there is a region of mixing above and below theintrusion, across which the density varies smoothly from that of the fluid below to that above theintruding flow. They proposed that this intermediate mixing zone could be described in termsof an effective turbulent mixing between the neutral and maximum height of the plume, withthe effective mixing diffusivity being proportional to the plume velocity and radius, and the ratioof the plume area to the overall area of the ambient fluid. They compared a series of numericalsolutions of the resultant equation

∂ρ

∂t+ πq

A∂ρ

∂z= α

∂

∂z

[ub3

A∂ρ

∂z

], (41)

where α is constant in the region between the neutral height and the top height of the plume, andu and b are the plume speed and radius at the neutral height with experimental data for the mixingof two-layer and continuously stratified ambient fluid, respectively. The model provides a good fitto the experimental data, both for the case of a two-layer ambient in which the plume is arrestedat the interface between the two layers and for the initial filling and subsequent deepening of themixed layer in the case of a continuously stratified fluid (Figure 9). The models suggest that if,in a stratified lake, a series of localized plumes penetrate down to the thermocline, then they willtend to mix the thermocline.

In the larger-scale oceanic context, with long-lived plumes, it is likely that the neutral cloudwill become geostrophically unstable and separate from the plume. Indeed, this is thought to bethe case with the convective line plumes that develop below long cracks in the ice cover, known asleads. The buoyancy flux associated with the cold, saline-rich water released as the surface waterrefreezes generates a descending line plume that is typically trapped at the thermocline in thepolar ocean. Here the spreading intrusion will break up into a series of geostrophically adjustededdies as discussed above (Bush & Woods 2000). The effect of rotation on the efficiency of themixing across a density interface has not been fully explored.


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Offset buoyancy (arbitrary units)

Hei

gh

t (ar

bit

rary

un

its)

0 2 4 6 8 10 12 14

0

1

2

3

4

5

6

7

8

Figure 9Evolution of the density profile as a function of time when a buoyant plume rises into a density-stratifiedlayer and mixes the layer. The profiles are offset in time to illustrate the mixing both back toward the sourceand upward into the stratified layer. Solid lines represent (dimensionless) experimental buoyancy profiles as afunction of (dimensionless) height obtained from a conductivity probe, whereas the dashed lines are obtainedfrom the solution of Equation 41 as described in Mott & Woods 2009a.

8.2. Filling Box Flows with Bubble Plumes

When a plume produced by a source of small bubbles with rise speed vs issues into the base of alayer of confined fluid of area A, then a balance can become established at a particular height atwhich the downward return flow associated with the plume, πq/A, where the plume has volumeflux πq, equals the rise speed of the bubbles. Above this level, there is a zone of bubble-ladenfluid, whereas below, where the bubble rise speed exceeds the return flow, the ambient fluid isbubble-free. As expected from plume theory, with a source buoyancy flux π f given in terms of thebubble volume flux Qb, and the density contrast with the water (ρω − ρb ),

π f = Qb g(ρω − ρb )/ρw, (42)

the equilibrium height is a distance z above the base of the tank, given by

vs ≈ 0.15(π f )1/3z5/3/A, (43)

where vs is the rise speed of the bubble (Phillips & Woods 2001). In basaltic magma chambers, theformation of such bubble-rich and bubble-poor layers of magma can affect the eruption dynamics,if there is a transition from eruption of the upper bubbly magma to the lower bubble-poor magma(Figure 10). Analogous effects can occur with a descending particle-laden plume, again if thesettling velocity is sufficiently small so as to be comparable with the return flow speed in theambient fluid.

8.3. Filling Boxes with Ventilation

There are numerous other phenomena related to the mixing of turbulent buoyant plumes inconfined spaces. Of particular note is the dynamics of naturally ventilated buildings, in whicha localized heat source produces a turbulent plume, which is coupled with a buoyancy-drivenventilation flow in which interior warm air vents from the top of the space, whereas cool externalair enters through an opening in the base of the building (Linden 1999). Linden et al. (1990)identified that, in steady state, the interior develops a two-layer stratification, with a lower layerof dense exterior fluid beneath an upper layer of buoyant fluid that the plume supplies from the

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Bubble rise =return flow

z

Bubbly layer

Upperbubbly layer

Original magma

Hybridmagma

First front

a b

Figure 10(a) Schematic of the stratification of a basaltic magma chamber owing to the separation of the bubbles fromthe magma. (b) Experimental model using electrolysis to produce a continuous source of small bubbles at thebase of a tank of water; the ensuing bubble plume leads to the formation of an upper bubble-laden layer, afterPhillips & Woods (2001).

mixture of hot source fluid and lower-layer fluid entrained by the plume. The transient dynamicsof these flow regimes resulting from an increase or decrease in the heat load has recently beenexplored (see Bower et al. 2008, Kaye & Hunt 2004).

However, we conclude with a simple plume-driven mixing flow that arises when an enclosedspace, filled with relatively buoyant fluid, exchanges fluid through an opening at a high levelwith an exterior of relatively dense fluid (e.g., a hot building or an underground mine in winter).Typically the dense inflow forms a turbulent descending plume in the interior of the space. Owingto the entrainment, this results in an upward return flow, with the interior fluid rising to the topof the space and then venting. Once the first front reaches the outflow vent, there is a transition inthe flow through the opening from the initial steady exchange flow associated with the constantdensity contrast in the vent, to a gradually waning exchange flow as progressively denser interiorfluid reaches the opening. Kuesters & Woods (2009) have shown experimentally and theoreticallythat in this regime, with the exchange flow Qe through the opening given in terms of the openingarea, A, and the buoyancy contrast across the opening, g′, as

Qe = c A5/4 g ′1/2, (44)

where c is the loss coefficient for the opening, then the interior density profile can be expressedin separable form ρ(z, t) = ρσ (z) ρη(t), where

ρη(t) = ρη(0)[1 + ωg 1/2

o c A5/4t/V]−2

. (45)

Here ω is an O(1) constant, which is found by solving for the vertical density profile, ρσ (z); V isthe volume of the space; and go is the initial buoyancy of the interior fluid. This process providesinsight into (a) the mixing in a confined ocean basin as driven by a dense exchange flow acrossa sill, which connects the basin to denser ocean water, although the model could be extended toaccount for the effects of rotation on such exchange flows, and also (b) the ventilation of a hot orsmoke-filled room, which only has a high-level opening connecting it to the cold exterior.


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9. SUMMARY AND OUTLOOK

Turbulent buoyant plumes, which are ubiquitous in nature, can transport large volumes of fluidthrough density stratified environments and can mix large volumes of fluid in enclosed spaces.Models of their dynamics, although highly simplified, enable us to infer some details of the mag-nitude or nature of the processes leading to buoyant plumes. In the volcanic context, both theheat flux and the particle load have a critical impact on the dynamics, whereas with hydrothermalplumes, both the composition and temperature fields are key to determining the source heat flux.There are many more complex processes for which plume theory has been or can be developed,including the effects of slip between phases (relevant for plumes with larger particles), and othermultiphase flow processes, including the effects of bubble coalescence and breakup. To conclude,we note that, as well as numerous natural applications, an interesting industrial challenge concernsthe application of natural plume-driven mixing for stirring up fluids in reactor vessels; this couldbe substantially more energy efficient than intensive propeller-driven mixing systems.

DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or financial holdings that mightbe perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

Colm Caulfield is thanked for providing a very thorough review of the material presented in thisarticle.

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AR400-FM ARI 13 November 2009 15:33

Annual Review ofFluid Mechanics

Volume 42, 2010Contents

Singular Perturbation Theory: A Viscous Flow out of GottingenRobert E. O’Malley Jr. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Dynamics of Winds and Currents Coupled to Surface WavesPeter P. Sullivan and James C. McWilliams � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �19

Fluvial Sedimentary PatternsG. Seminara � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �43

Shear Bands in Matter with GranularityPeter Schall and Martin van Hecke � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �67

Slip on Superhydrophobic SurfacesJonathan P. Rothstein � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �89

Turbulent Dispersed Multiphase FlowS. Balachandar and John K. Eaton � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 111

Turbidity Currents and Their DepositsEckart Meiburg and Ben Kneller � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 135

Measurement of the Velocity Gradient Tensor in Turbulent FlowsJames M. Wallace and Petar V. Vukoslavcevic � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 157

Friction Drag Reduction of External Flows with Bubble andGas InjectionSteven L. Ceccio � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 183

Wave–Vortex Interactions in Fluids and SuperfluidsOliver Buhler � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 205

Laminar, Transitional, and Turbulent Flows in Rotor-Stator CavitiesBrian Launder, Sebastien Poncet, and Eric Serre � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 229

Scale-Dependent Models for Atmospheric FlowsRupert Klein � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 249

Spike-Type Compressor Stall Inception, Detection, and ControlC.S. Tan, I. Day, S. Morris, and A. Wadia � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 275

vii

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AR400-FM ARI 13 November 2009 15:33

Airflow and Particle Transport in the Human Respiratory SystemC. Kleinstreuer and Z. Zhang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 301

Small-Scale Properties of Turbulent Rayleigh-Benard ConvectionDetlef Lohse and Ke-Qing Xia � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 335

Fluid Dynamics of Urban Atmospheres in Complex TerrainH.J.S. Fernando � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 365

Turbulent Plumes in NatureAndrew W. Woods � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 391

Fluid Mechanics of MicrorheologyTodd M. Squires and Thomas G. Mason � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 413

Lattice-Boltzmann Method for Complex FlowsCyrus K. Aidun and Jonathan R. Clausen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 439

Wavelet Methods in Computational Fluid DynamicsKai Schneider and Oleg V. Vasilyev � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 473

Dielectric Barrier Discharge Plasma Actuators for Flow ControlThomas C. Corke, C. Lon Enloe, and Stephen P. Wilkinson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 505

Applications of Holography in Fluid Mechanics and Particle DynamicsJoseph Katz and Jian Sheng � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 531

Recent Advances in Micro-Particle Image VelocimetrySteven T. Wereley and Carl D. Meinhart � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 557

Indexes

Cumulative Index of Contributing Authors, Volumes 1–42 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 577

Cumulative Index of Chapter Titles, Volumes 1–42 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 585

Errata

An online log of corrections to Annual Review of Fluid Mechanics articles may be foundat http://fluid.annualreviews.org/errata.shtml

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Turbulent Plumes in Nature Click here for quick links to Annual Reviews content online, including:...

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Transcript of Turbulent Plumes in Nature Click here for quick links to Annual Reviews content online, including:...